Rotational or Angular Motion Created by C. Wells. This is available for use by students and teachers but is not authorized for reproduction. Please inform author of any errrors.

When an object’s center of mass moves from one place to another, we call it linear motion or translational motion.

Rotational motion (also called angular motion) describes the motion of an object around a fixed line called the “axis of rotation” or around a fixed point called the “fulcrum” or “pivot”.

Suppose we have a wheel that is attached to wall at its center so that it can spin (sort of like the “Wheel of Fortune”) around an axle.

Will a force cause it to spin? This force will NOT cause it to spin…because it is directed through the pivot or axle.

Will a force directed inward on the wheel cause it to spin? Push straight into the page on the center. No, a force straight inward on the pivot (green dot) will NOT cause it to spin, because the force is directed through the pivot.

So…what will cause the wheel to spin? R F If we direct the force so that it does NOT go through the pivot, then it WILL cause the wheel to spin. This force is a distance, R, from the center. We call this distance the “lever arm”…..and we call this type of force a torque.

An object will only rotate (or spin) if a torque is applied to it An object will only rotate (or spin) if a torque is applied to it. A torque is a force that does NOT go through the axle or pivot. Therefore, a torque has a lever arm:  = FR

To calculate the torque, multiply the force applied times the perpendicular distance from the line of force to the pivot.

Suppose we apply a force of 10 newtons to the edge of the wheel, which has a radius of 0.5 meters. What is the size of the torque? R F Answer: Since the force and radius are already perpendicular, the torque is just the product of the two.  = (10 N)(0.5 m) = 5 N-m

Now, calculate the torque when the force is applied at a 30 degree angle, as shown below. 30 degree angle from tangent Answer: The force and lever arm are not perpendicular, so we have to use the component of the force that is perpendicular.  = (10 cos 30)(.5)

Use the right hand rule to determine the direction of torque.

When something simply rotates or spins (like our wheel attached to a wall), it isn’t going anywhere, so it has no linear velocity. Instead, we describe how fast it is spinning with its angular velocity. Watch the spinning girl at http://www.theness.com/neurologicablog/?p=27

Angular velocity tells how far something turns in a certain amount of time. The amount it turns is an angle or angular displacement.

Let’s consider the Earth: (a) What is the magnitude of the angular velocity (or angular speed) of the Earth as it spins on it axis? (b) What is the direction of the Earth’s angular velocity?

Answers:

As we look at this clock face: What is the angular velocity of the hour hand? What is the angular velocity of the minute hand? What is the angular velocity of the second hand? What is the direction of the torque the clock motor applies to make these hands move?

The mass of an object helps us to describe the amount of force it will take it to start it in linear motion or to stop it. However, mass alone isn’t enough to help us describe rotational motion.

To describe how an object will rotate… To describe how an object will rotate….and the amount of torque required to start it or stop it….we need to know the mass of the object and how far from the pivot the mass is concentrated. We call this the moment of inertia or rotational inertia of the object.

Moment of inertia is mass times the radius of the object squared times a coefficient that describes how far from the center the mass is concentrated. An object has that moment of inertia whether it is spinning or stationary. For example: solid ball: I = 2/5 mr2 hollow ball: I = 2/3 mr2

Objects that are spinning have angular momentum, L, that depends upon the moment of inertia of the object and how fast it is spinning (angular velocity): L = I Use the right hand rule to determine the direction of angular velocity and angular momentum---which are in the same direction.

When this skater brings in his arms, he begins to spin faster. He has decreased his radius of rotation, so his moment of inertia is decreased. By the law of conservation of angular momentum, a decrease in moment of inertia means an increase in angular speed. www.youtube.com [Search “one foot spin”]

Torque R F

Remember that torque, , is the product of force and lever arm Remember that torque, , is the product of force and lever arm. Another way of stating this is that “torque is the applied force times the perpendicular distance from the line of force to the pivot”.

If the green board is balanced at its center on the fulcrum and a force, F, is applied as shown, then the distance L is the “lever arm”, since it is the perpendicular distance from the line of the force to the pivot or fulcrum. F L The force produces a torque that is equal to F times L and produces a clockwise rotation of the board.

Now, if a another force that is equal to the first is applied at the same distance on the other side of the pivot, it produces an equal amount of torque but in the opposite direction. F L The net torque now adds to zero—and the board does not rotate. The board is in rotational equilibrium. Note: This will only be true if the board is uniform and the pivot is at the center of the board, so that the gravitational force is causing no torque on the board.

Objects are in static equilibrium iff: 1 Objects are in static equilibrium iff: 1. The net force in the x-direction is zero. 2. The net force in the y-direction is zero. 3. The net torque is zero.

Problems assigned in Chapter 8: 3, 23, 25 Problems assigned in Chapter 9: 3, 5, 9, 11,15, 20, 21, 26, 27